↑

Properties

Label 117.2.e.c
Level $117$
Weight $2$
Character orbit 117.e
Analytic conductor $0.934$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(40,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.40");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.e (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 3 x^{10} - x^{9} - 2 x^{8} + 9 x^{7} + 24 x^{6} + 27 x^{5} - 18 x^{4} - 27 x^{3} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{2}+ \cdots + (2 \beta_{11} - 2 \beta_{10} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{2}+ \cdots + (\beta_{9} - 3 \beta_{8} - 3 \beta_{7} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{3} - 6 q^{4} + 3 q^{5} - 7 q^{6} - 12 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 2 q^{3} - 6 q^{4} + 3 q^{5} - 7 q^{6} - 12 q^{8} - 2 q^{9} - 12 q^{10} + 7 q^{11} + 7 q^{12} + 6 q^{13} + 13 q^{14} - 12 q^{15} - 6 q^{16} - 28 q^{17} + 26 q^{18} - 6 q^{19} + 17 q^{20} - 18 q^{21} + 3 q^{22} + 17 q^{23} - 30 q^{24} - 3 q^{25} + 4 q^{26} + 13 q^{27} + 30 q^{28} + 14 q^{29} - 7 q^{30} - 6 q^{31} + 9 q^{32} + 19 q^{33} - 3 q^{34} - 26 q^{35} + 43 q^{36} - 18 q^{37} - 4 q^{38} - q^{39} + 6 q^{40} - 4 q^{41} - 17 q^{42} + 3 q^{43} - 26 q^{44} + 18 q^{45} + 9 q^{47} - 3 q^{48} - 21 q^{50} - 4 q^{51} + 6 q^{52} - 56 q^{53} + 35 q^{54} + 30 q^{55} + 16 q^{56} - 30 q^{57} - 18 q^{58} + 17 q^{59} - 31 q^{60} + 6 q^{61} + 8 q^{62} - 60 q^{64} - 3 q^{65} - 19 q^{66} + 12 q^{67} + 22 q^{68} - 13 q^{69} - 9 q^{70} + 2 q^{71} - 18 q^{73} - 27 q^{74} + 18 q^{75} + 3 q^{76} + 15 q^{77} - 2 q^{78} + 6 q^{79} + 48 q^{80} - 2 q^{81} + 54 q^{82} + 28 q^{83} + 31 q^{84} - 27 q^{85} - 6 q^{86} + 23 q^{87} + 9 q^{88} + 18 q^{89} + 26 q^{90} + 44 q^{92} + 2 q^{93} + 24 q^{94} + 32 q^{95} - 57 q^{96} - 34 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 3 x^{10} - x^{9} - 2 x^{8} + 9 x^{7} + 24 x^{6} + 27 x^{5} - 18 x^{4} - 27 x^{3} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + \nu^{10} + 3 \nu^{9} + \nu^{8} + 2 \nu^{7} - 9 \nu^{6} - 24 \nu^{5} - 27 \nu^{4} + \cdots + 243 ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7 \nu^{11} + 13 \nu^{10} + 60 \nu^{9} + 79 \nu^{8} + 62 \nu^{7} - 120 \nu^{6} - 582 \nu^{5} + \cdots + 7533 ) / 729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + 12 \nu^{10} + 31 \nu^{9} + 28 \nu^{8} - 133 \nu^{6} - 345 \nu^{5} - 456 \nu^{4} + \cdots + 3321 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10 \nu^{11} + 14 \nu^{10} - 28 \nu^{8} - 98 \nu^{7} - 147 \nu^{6} - 84 \nu^{5} + 252 \nu^{4} + \cdots - 2187 ) / 243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 103 \nu^{11} + 116 \nu^{10} - 87 \nu^{9} - 364 \nu^{8} - 1046 \nu^{7} - 1329 \nu^{6} + 51 \nu^{5} + \cdots - 32319 ) / 1458 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 133 \nu^{11} - 176 \nu^{10} + 51 \nu^{9} + 394 \nu^{8} + 1358 \nu^{7} + 1941 \nu^{6} + \cdots + 33777 ) / 1458 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24 \nu^{11} + 17 \nu^{10} - 47 \nu^{9} - 105 \nu^{8} - 233 \nu^{7} - 175 \nu^{6} + 300 \nu^{5} + \cdots - 10044 ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73 \nu^{11} + 74 \nu^{10} - 87 \nu^{9} - 280 \nu^{8} - 752 \nu^{7} - 807 \nu^{6} + 303 \nu^{5} + \cdots - 25029 ) / 486 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 103 \nu^{11} - 66 \nu^{10} + 211 \nu^{9} + 472 \nu^{8} + 1014 \nu^{7} + 731 \nu^{6} - 1353 \nu^{5} + \cdots + 43659 ) / 486 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65 \nu^{11} - 34 \nu^{10} + 150 \nu^{9} + 317 \nu^{8} + 634 \nu^{7} + 378 \nu^{6} - 1083 \nu^{5} + \cdots + 29403 ) / 243 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{9} - 2\beta_{5} + \beta_{4} + 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + 3\beta_{10} - \beta_{9} - 4\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{7} + 4\beta_{6} + 3\beta_{5} - 3\beta_{4} + 4\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} + 3 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8 \beta_{11} + 3 \beta_{10} - \beta_{9} - 4 \beta_{7} - 19 \beta_{6} - 4 \beta_{5} + 5 \beta_{4} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{11} + 6 \beta_{10} - 31 \beta_{9} + 18 \beta_{8} - 5 \beta_{7} + 16 \beta_{6} - 9 \beta_{5} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 20 \beta_{11} + 3 \beta_{10} + 8 \beta_{9} + 12 \beta_{8} - 18 \beta_{7} + 27 \beta_{6} - 5 \beta_{5} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 32 \beta_{11} + 39 \beta_{10} + 5 \beta_{9} - 36 \beta_{8} + 8 \beta_{7} + 29 \beta_{6} + 38 \beta_{5} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 54 \beta_{11} - 57 \beta_{10} + 11 \beta_{9} + 153 \beta_{8} + 133 \beta_{7} - 35 \beta_{6} + 24 \beta_{5} + \cdots + 62 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 94 \beta_{11} + 111 \beta_{10} - 85 \beta_{9} + 21 \beta_{8} - 78 \beta_{7} + 12 \beta_{6} + \cdots + 45 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−1.29386 − 1.15149i
−0.372202 + 1.69159i
1.70010 − 0.331167i
0.471837 − 1.66654i
1.70358 + 0.312736i
−1.70946 + 0.278853i
−1.29386 + 1.15149i
−0.372202 − 1.69159i
1.70010 + 0.331167i
0.471837 + 1.66654i
1.70358 − 0.312736i
−1.70946 − 0.278853i
−1.10275 − 1.91002i −0.350291 − 1.69626i −1.43212 + 2.48050i 1.80370 − 3.12410i −2.85361 + 2.53961i 0.540062 + 0.935414i 1.90608 −2.75459 + 1.18837i −7.95614
40.2 −0.636681 − 1.10276i 1.65106 + 0.523457i 0.189274 − 0.327833i 0.134264 − 0.232553i −0.473948 − 2.15400i 0.577621 + 1.00047i −3.02875 2.45199 + 1.72852i −0.341935
40.3 0.129090 + 0.223591i −1.13685 + 1.30674i 0.966671 − 1.67432i −1.73641 + 3.00755i −0.438932 − 0.0855008i 2.21165 + 3.83070i 1.01551 −0.415158 − 2.97114i −0.896615
40.4 0.248047 + 0.429630i −1.67919 − 0.424649i 0.876945 − 1.51891i 0.663441 − 1.14911i −0.234075 − 0.826762i −1.56464 − 2.71004i 1.86228 2.63935 + 1.42613i 0.658258
40.5 1.10985 + 1.92231i −0.580954 + 1.63171i −1.46353 + 2.53491i 1.13044 − 1.95798i −3.78144 + 0.694180i −0.183027 − 0.317013i −2.05779 −2.32499 − 1.89590i 5.01848
40.6 1.25245 + 2.16930i 1.09622 − 1.34101i −2.13724 + 3.70181i −0.495441 + 0.858128i 4.28200 + 0.698496i −1.58167 − 2.73953i −5.69734 −0.596596 − 2.94008i −2.48205
79.1 −1.10275 + 1.91002i −0.350291 + 1.69626i −1.43212 − 2.48050i 1.80370 + 3.12410i −2.85361 − 2.53961i 0.540062 − 0.935414i 1.90608 −2.75459 − 1.18837i −7.95614
79.2 −0.636681 + 1.10276i 1.65106 − 0.523457i 0.189274 + 0.327833i 0.134264 + 0.232553i −0.473948 + 2.15400i 0.577621 − 1.00047i −3.02875 2.45199 − 1.72852i −0.341935
79.3 0.129090 − 0.223591i −1.13685 − 1.30674i 0.966671 + 1.67432i −1.73641 − 3.00755i −0.438932 + 0.0855008i 2.21165 − 3.83070i 1.01551 −0.415158 + 2.97114i −0.896615
79.4 0.248047 − 0.429630i −1.67919 + 0.424649i 0.876945 + 1.51891i 0.663441 + 1.14911i −0.234075 + 0.826762i −1.56464 + 2.71004i 1.86228 2.63935 − 1.42613i 0.658258
79.5 1.10985 − 1.92231i −0.580954 − 1.63171i −1.46353 − 2.53491i 1.13044 + 1.95798i −3.78144 − 0.694180i −0.183027 + 0.317013i −2.05779 −2.32499 + 1.89590i 5.01848
79.6 1.25245 − 2.16930i 1.09622 + 1.34101i −2.13724 − 3.70181i −0.495441 − 0.858128i 4.28200 − 0.698496i −1.58167 + 2.73953i −5.69734 −0.596596 + 2.94008i −2.48205
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.e.c ✓ 12
3.b odd 2 1 351.2.e.c 12
9.c even 3 1 inner 117.2.e.c ✓ 12
9.c even 3 1 1053.2.a.l 6
9.d odd 6 1 351.2.e.c 12
9.d odd 6 1 1053.2.a.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.e.c ✓ 12 1.a even 1 1 trivial
117.2.e.c ✓ 12 9.c even 3 1 inner
351.2.e.c 12 3.b odd 2 1
351.2.e.c 12 9.d odd 6 1
1053.2.a.l 6 9.c even 3 1
1053.2.a.m 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 2 T_{2}^{11} + 11 T_{2}^{10} - 10 T_{2}^{9} + 63 T_{2}^{8} - 55 T_{2}^{7} + 196 T_{2}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 100 \) Copy content Toggle raw display
$7$ \( T^{12} + 21 T^{10} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{12} - 7 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$17$ \( (T^{6} + 14 T^{5} + 62 T^{4} + \cdots + 5)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 3 T^{5} + \cdots + 196)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 17 T^{11} + \cdots + 4669921 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 868480900 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + \cdots + 91204 \) Copy content Toggle raw display
$37$ \( (T^{6} + 9 T^{5} + \cdots + 16562)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 802588900 \) Copy content Toggle raw display
$43$ \( T^{12} - 3 T^{11} + \cdots + 5013121 \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 5476 \) Copy content Toggle raw display
$53$ \( (T^{6} + 28 T^{5} + \cdots - 6983)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 29191089316 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 312829969 \) Copy content Toggle raw display
$67$ \( T^{12} - 12 T^{11} + \cdots + 6697744 \) Copy content Toggle raw display
$71$ \( (T^{6} - T^{5} + \cdots - 195956)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 9 T^{5} + \cdots + 5782)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 6 T^{11} + \cdots + 23049601 \) Copy content Toggle raw display
$83$ \( T^{12} - 28 T^{11} + \cdots + 32330596 \) Copy content Toggle raw display
$89$ \( (T^{6} - 9 T^{5} + \cdots + 78148)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 228 T^{10} + \cdots + 59536 \) Copy content Toggle raw display
show more
show less